Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $z = \dfrac{p^2 + 4p}{p^2 + 5p + 4} \div \dfrac{p^2 - 3p}{p^2 + 6p + 5} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{p^2 + 4p}{p^2 + 5p + 4} \times \dfrac{p^2 + 6p + 5}{p^2 - 3p} $ First factor out any common factors. $z = \dfrac{p(p + 4)}{p^2 + 5p + 4} \times \dfrac{p^2 + 6p + 5}{p(p - 3)} $ Then factor the quadratic expressions. $z = \dfrac {p(p + 4)} {(p + 1)(p + 4)} \times \dfrac {(p + 1)(p + 5)} {p(p - 3)} $ Then multiply the two numerators and multiply the two denominators. $z = \dfrac {p(p + 4) \times (p + 1)(p + 5) } { (p + 1)(p + 4) \times p(p - 3)} $ $z = \dfrac {p(p + 1)(p + 5)(p + 4)} {p(p + 1)(p + 4)(p - 3)} $ Notice that $(p + 1)$ and $(p + 4)$ appear in both the numerator and denominator so we can cancel them. $z = \dfrac {p\cancel{(p + 1)}(p + 5)(p + 4)} {p\cancel{(p + 1)}(p + 4)(p - 3)} $ We are dividing by $p + 1$ , so $p + 1 \neq 0$ Therefore, $p \neq -1$ $z = \dfrac {p\cancel{(p + 1)}(p + 5)\cancel{(p + 4)}} {p\cancel{(p + 1)}\cancel{(p + 4)}(p - 3)} $ We are dividing by $p + 4$ , so $p + 4 \neq 0$ Therefore, $p \neq -4$ $z = \dfrac {p(p + 5)} {p(p - 3)} $ $ z = \dfrac{p + 5}{p - 3}; p \neq -1; p \neq -4 $